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In this paper, we investigate the existence of concave and monotone positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form \(x^{(4)}(t)=f(t,x(t),x'(t),x''(t))\), \(t\in[0,1]\), \(x(0)=x'(1)=x'''(1)=0\), \(x''(0)=\int_{0}^{1}g(s)x''(s) \, \mathrm{d}s\), where \(f\in C([0,1]\times[0,+\infty)^{2}\times(-\infty,0],[0,+\infty))\), \(g\in C([0,1],[0,+\infty))\). By using a fixed point theorem of cone expansion and compression of norm type, the existence and nonexistence of concave and monotone positive solutions for the above boundary value problems is obtained. Meanwhile, as applications of our results, some examples are given.

Monotone positive solution of a fourth-order BVP with integral boundary conditions